Multiplication Tables

A very important foundation for understanding math

Sometimes I see students who are really intelligent but who are having problems at school understanding new material. Some of these students only need to go over a few specific examples to grasp the concepts and move on to the next topic. So it is sad and almost unbelievable to discover that, in a few cases, the real obstacle standing on their way is that they do not know the multiplication tables! I remember when I was in second grade I hated learning the multiplication tables because the repetition process was so boring and it seemed meaningless to me at the time. However, in third grade I discovered the benefits of knowing by heart the multiplication tables. It allowed me to understand division. Understanding division allowed me to build a solid understanding of fractions.
In this time and age, many generations have grown up and gone through school using pocket calculators. A few people have made it all the way to college without ever learning how to multiply two numbers without using a calculator. The problem for them is, the more advanced the math courses they take, the more trouble they have at trying to figure out how formulas work by looking at specific examples. They cannot think their way through the examples because they don't know their multiplication tables; therefore their mastery of division, fractions and exponents is very limited and shaky.
To students who are preparing for the GMAT, GRE or CBEST, I always recommend to review, polish, extend and reinforce their knowledge of multiplication tables. The importance of this foundation cannot be stressed enough.

Absolute value expressions in 5th grade?

Too abstract topics too early

One of my students is in 5th grade. Not long ago his homework consisted of writing down the full-blown English names of twelve-digit numbers, like 535,176,402,988. He kept busy writing line after line of tens of millions, and hundreds of billions. After a while, the assignment seemed pretty boring to me but my student was interested in the task all the way through. I think a big reason for his motivation was that he was able to do it. The big numbers seemed challenging to him, but the task was doable because he completely understood the principles involved in the translation.
Last week his parents asked me to go with him over some questions he got wrong in a quiz. I was amazed to find in this quiz questions involving absolute value expressions!
I was like: Absolute value in 5th grade? What for?
I don’t know about you but it does not make any sense to me. I mean, the first time I knew absolute value existed, I was in 12th grade, at the end of high school. Now they are covering absolute value in elementary school? Give me a break!
It was kind of hard to explain his mistakes to him, in part because the absolute value concept is way more abstract than the concept of hundreds of millions, and in part because he did not want to accept he made a mistake. So he was ecstatic when I discovered that in one of the three problems he was marked down he actually had selected the right answer. He was right on that particular problem, not wrong.
Which kind of proves my point, in a way. The absolute value concept is too abstract not only for most 5th grade students, but apparently for some 5th grade teachers as well.

You Don't Have To Do Anything

Think about what you can do, not what you "have to" do

Very often students seem to freeze when they see some kind of problem. In this situations I ask: "What are you thinking? What is going through your mind right now? What thoughts, feelings or ideas do you have when you read this problem?" By asking this types of questions repeatedly, I have discovered that, in many cases, when students find a particular kind of problem (the exact type varies from student to student), they think they are supposed to follow some steps, some fixed routine they were taught at some point in the past by one of their teachers. The problem is now they don't remember what are those steps they think they have to follow, and more importantly, in most cases they never totally understood the reasons why those steps work.
They typically give me answers like:
"Well, I think I have to multiply these two numbers, but I am not sure..."
"Ah, I need to add these fractions, but I don't know how..."
"I forgot what formula I have to use for this problem..."
"I am supposed to set up an equation, right? But how?"
The key words these answers have in common are verbal forms like: "I have to," "I need to," "I am supposed to," etcetera. They really think there is something very specific they have to do, and they just don't know what that is.
When I spot this blockage I say: "You don't have to do anything!" They look at me and they go: "I don't?" They look quite surprised but relieved at the same time. Then I say:
"No, you don't have to do anything. You don't even have to solve the problem. I mean, you want to solve the problem because you want to pass your test, right? But you don't have to, you want to. Now, to solve the problem, you can do that by going whatever way it works. Nobody is going to be looking over your shoulder to see how you do it. It's a multiple choice question. Nobody cares how you do it. The only thing that counts is whether your final answer is right or wrong, right?"
After I make my point clear, they usually ask: "But, then, how can I solve it? What can I do?" And I say: "Exactly! Perfect! That is the right question. What can you do? Well, what do we have? Look at the problem, look at the numbers, look at those expressions. What can we do?" Then all of a sudden they go: "Oh! I can take the 50 out on both sides, then I can substitute this variable for that other formula!" or whatever the case may be, but they start working their way to the solution.
The words we choose to talk to ourselves make a big difference. When students think in terms of "I have to," "I need to," "I am supposed to," those words take them to a mental and emotional state where they were just going through the motions and mechanically repeated meaningless tasks they didn't understand and they didn't care about.
If instead they think about their possibilities, their options, about what they can do, then it's much more likely they will find the clarity, creativity and initiative that will lead them to finding the right solution by themselves.